Bücher der Reihe Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory.

This book introduces the mathematical methods of theoretical and experimental quantum field theory, emphasizing coordinate-free presentations of the objects in play. Offers examples of classical field theories, discusses renormalization methods and more.

Rio de Janeiro, January 1987 Ricardo Mane Introduction This book is an introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems, which is sometimes called differentiable ergodic theory.

Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec tures. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils.

The spectra of transfer operators associated to dynamical systems, when acting on suitable Banach spaces, contain key information about the ergodic properties of the systems. Focusing on expanding and hyperbolic maps, this book gives a self-contained account on the relation between zeroes of dynamical determinants, poles of dynamical zeta functions, and the discrete spectra of the transfer operators.In the hyperbolic case, the first key step consists in constructing a suitable Banach space of anisotropic distributions. The first part of the book is devoted to the easier case of expanding endomorphisms, showing how the (isotropic) function spaces relevant there can be studied via Paley¿Littlewood decompositions, and allowing easier access to the construction of the anisotropic spaces which is performed in the second part.This is the first book describing the use of anisotropic spaces in dynamics. Aimed at researchers and graduate students, it presents results and techniques developed since the beginning of the twenty-first century.

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field.

In the 1980s, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". It requires no knowledge whatsoever of any physics, and contains proofs in complete detail of much of what is rigorously known on spin glasses at the time of writing.

In the case of completely integrable systems, periodic solutions are found by inspection.

The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces.

This is a revised, updated and enlarged edition of Volume II of the author's book "Spin Glasses: A Challenge for Mathematicians" designed to introduce this exciting work to the math-minded reader, in a rigorous manner requiring no knowledge of any physics.

This self-contained introduction treats thin geometries and thick buildings from a diagrammatic perspective, covers polar geometries whose projective planes are Desarguesian and offers a basic reference for study of diagram geometry. Includes many examples.

John Hornstein has written about the author's theorem on nonintegrability of geodesic flows on closed surfaces of genus greater than one: "Here is an example of how differential geometry, differential and algebraic topology, and Newton's laws make music together" (Amer.

This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Introduction . . . . . . . . Basic Properties of n-Widths . Properties of d * * * * * * * * * * 9 n 15 2. Existence of Optimal Subspaces for d * n n 17 3. Properties of d * * * * * * 20 4. Properties of b * * * * * * n 5. Duality Between d and d * * 27 n 7. Some Relationships Between dn(T), dn(T) and bn(T) . 32 37 Notes and References . . . . .

This book covers the basics of Clifford algebras and spinor modules, with applications to the theory of Lie groups. Topics include Petracci's proof of the Poincare-Birkhoff-Witt theorem, quantized Weil algebras, Duflo's theorem for quadratic Lie algebras and more.

This book presents a comprehensive, encyclopedic approach to the subject of foliations, one of the major concepts of modern geometry and topology. It addresses graduate students and researchers and serves as a reference book for experts in the field.

The theme of this book is an examination of the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds, offering the first complete account of Oka-Grauert theory and its modern extensions.

The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver.

Field Arithmetic explores Diophantine fields through their absolute Galois groups.

A quadratic differential on aRiemann surface is locally represented by a ho lomorphic function element wh ich transforms like the square of a derivative under a conformal change of the parameter. A quadratic differential defines, in a natural way, a field of line elements on the surface, with singularities at the critical points, i.e.

Neron models were invented by A. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Neron models from the point of view of Grothendieck's algebraic geometry.

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem.